Logic puzzle: find the real date of birth from several options

Albert does not know Cheryl's birthday as there are several possible choices in each month. But from his knowledge of the month, he can extract information and convey it to Bernard.

Albert says Bernard doesn't know the exact date. How does he know this? From my knowledge of the month. Albert argues as follows: if Cheryl told Bernard the number 18 or 19, he would immediately name the month, since the list contains exactly one date with such numbers.

Bernard understands this and concludes that Sherrill could not name Alberta either May or June (the months in which there are numbers 18 and 19). This means that the required month is either July or August.

Initially, Bernard did not know when Cheryl's birthday was. But after the remark Albert guessed. How? Of the dates remaining in July and August, only 14 occur twice. If Cheryl told Bernard this number, then after Albert's guess, he still could not give an exact answer. The fact that he immediately understood everything suggests that Cheryl was not born on the 14th. There are three possible dates left: July 16, August 15 and August 17.

After Bernard reported his success to Albert, he also realized when Cheryl birthday. He could do this only if the month that was named to him is July. Otherwise, he could not give an exact answer and would hesitate between two possible dates - August 15 and August 17. It follows that Cheryl was born on July 16th.